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The combination of configuration interaction and many-body perturbationtheory methods (CI+MBPT) is extended to non-perturbatively includeconfigurations with electron holes below the designated Fermi level, allowingus to treat systems where holes play an important role. For example, the methodcan treat valence-hole systems like Ir$^{17+}$, particle-hole excitations innoble gases, and difficult transitions such as the $6s \rightarrow 5d^{-1}6s^2$optical clock transition in Hg$^+$. We take the latter system as our test casefor the method and obtain very good accuracy (~1%) for the low-lying transitionenergies. The $\alpha$-dependence of these transitions is calculated and usedto reinterpret the existing best laboratory limits on the time-dependence ofthe fine-structure constant.

Particle-hole configuration interaction and many-body perturbation theory: application to Hg+